Abstract

Microlenses are highly attractive for optical applications such as super resolution and photonic nanojets, but their design is more demanding than the one of larger lenses because resonance effects play an important role, which forces one to perform a full wave analysis. Although mostly spherical microlenses were studied in the past, they may have various shapes and their optimization is highly demanding, especially, when the shape is described with many parameters. We first outline a very powerful mathematical tool: shape optimization based on shape gradient computations. This procedure may be applied with much less numerical cost than traditional optimizers, especially when the number of parameters describing the shape goes to infinity. In order to demonstrate the concept, we optimize microlenses using shape optimization starting from more or less reasonable elliptical and semi-circular shapes. We show that strong increases of the performance of the lenses may be obtained for any reasonable value of the refraction index.

© 2015 Optical Society of America

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References

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  1. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12, 1214–1220 (2004).
    [Crossref] [PubMed]
  2. M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express 19, 10206–10220 (2011).
    [Crossref] [PubMed]
  3. Y. Shen, L. V. Wang, and J.-T. Shen, “Ultralong photonic nanojet formed by a two-layer dielectric microsphere,” Opt. Lett. 39, 4120–4123 (2014).
    [Crossref] [PubMed]
  4. C. Hafner, “Boundary methods for optical nano structures,” physica status solidi (b) 244, 3435–3447 (2007).
    [Crossref]
  5. R. Hiptmair, A. Paganini, and S. Sargheini, “Comparison of approximate shape gradients,” BIT Numerical Mathematics, 1–27 (2014).
  6. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer, 2013).
    [Crossref]
  7. D. Braess, Finite elements. Theory, Fast Solvers, and Applications in Elasticity Theory (Cambridge University, 2007).
    [Crossref]
  8. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [Crossref]
  9. G. Allaire, Conception Optimale de Structures (Springer-Verlag, 2007).
  10. K. Höllig and J. Hörner, Approximation and modeling with B-splines (Society for Industrial and Applied Mathematics, 2013).
  11. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints (Springer, 2009).
  12. R. Hiptmair and A. Paganini, “Shape optimization by pursuing diffeomorphisms,” SAM-Report 2014-27, ETHZ (2015).
  13. J. Nocedal and S. J. Wright, Numerical Optimization (Springer, 2006).
  14. K. Eppler and H. Harbrecht, “Coupling of FEM and BEM in shape optimization,” Numer. Math. 104, 47–68 (2006).
    [Crossref]

2014 (1)

2011 (1)

2007 (1)

C. Hafner, “Boundary methods for optical nano structures,” physica status solidi (b) 244, 3435–3447 (2007).
[Crossref]

2006 (1)

K. Eppler and H. Harbrecht, “Coupling of FEM and BEM in shape optimization,” Numer. Math. 104, 47–68 (2006).
[Crossref]

2004 (1)

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

Allaire, G.

G. Allaire, Conception Optimale de Structures (Springer-Verlag, 2007).

Backman, V.

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

Braess, D.

D. Braess, Finite elements. Theory, Fast Solvers, and Applications in Elasticity Theory (Cambridge University, 2007).
[Crossref]

Chen, Z.

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer, 2013).
[Crossref]

Eppler, K.

K. Eppler and H. Harbrecht, “Coupling of FEM and BEM in shape optimization,” Numer. Math. 104, 47–68 (2006).
[Crossref]

Hafner, C.

C. Hafner, “Boundary methods for optical nano structures,” physica status solidi (b) 244, 3435–3447 (2007).
[Crossref]

Harbrecht, H.

K. Eppler and H. Harbrecht, “Coupling of FEM and BEM in shape optimization,” Numer. Math. 104, 47–68 (2006).
[Crossref]

Herzig, H. P.

Hinze, M.

M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints (Springer, 2009).

Hiptmair, R.

R. Hiptmair and A. Paganini, “Shape optimization by pursuing diffeomorphisms,” SAM-Report 2014-27, ETHZ (2015).

R. Hiptmair, A. Paganini, and S. Sargheini, “Comparison of approximate shape gradients,” BIT Numerical Mathematics, 1–27 (2014).

Höllig, K.

K. Höllig and J. Hörner, Approximation and modeling with B-splines (Society for Industrial and Applied Mathematics, 2013).

Hörner, J.

K. Höllig and J. Hörner, Approximation and modeling with B-splines (Society for Industrial and Applied Mathematics, 2013).

Kim, M.-S.

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer, 2013).
[Crossref]

Mühlig, S.

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer, 2006).

Paganini, A.

R. Hiptmair, A. Paganini, and S. Sargheini, “Comparison of approximate shape gradients,” BIT Numerical Mathematics, 1–27 (2014).

R. Hiptmair and A. Paganini, “Shape optimization by pursuing diffeomorphisms,” SAM-Report 2014-27, ETHZ (2015).

Pinnau, R.

M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints (Springer, 2009).

Rockstuhl, C.

Sargheini, S.

R. Hiptmair, A. Paganini, and S. Sargheini, “Comparison of approximate shape gradients,” BIT Numerical Mathematics, 1–27 (2014).

Scharf, T.

Shen, J.-T.

Shen, Y.

Taflove, A.

Ulbrich, M.

M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints (Springer, 2009).

Ulbrich, S.

M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints (Springer, 2009).

Wang, L. V.

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer, 2006).

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

Numer. Math. (1)

K. Eppler and H. Harbrecht, “Coupling of FEM and BEM in shape optimization,” Numer. Math. 104, 47–68 (2006).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

physica status solidi (b) (1)

C. Hafner, “Boundary methods for optical nano structures,” physica status solidi (b) 244, 3435–3447 (2007).
[Crossref]

Other (8)

R. Hiptmair, A. Paganini, and S. Sargheini, “Comparison of approximate shape gradients,” BIT Numerical Mathematics, 1–27 (2014).

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer, 2013).
[Crossref]

D. Braess, Finite elements. Theory, Fast Solvers, and Applications in Elasticity Theory (Cambridge University, 2007).
[Crossref]

G. Allaire, Conception Optimale de Structures (Springer-Verlag, 2007).

K. Höllig and J. Hörner, Approximation and modeling with B-splines (Society for Industrial and Applied Mathematics, 2013).

M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints (Springer, 2009).

R. Hiptmair and A. Paganini, “Shape optimization by pursuing diffeomorphisms,” SAM-Report 2014-27, ETHZ (2015).

J. Nocedal and S. J. Wright, Numerical Optimization (Springer, 2006).

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Figures (5)

Fig. 1
Fig. 1 A dielectric lens DL is illuminated by a plane wave Ein. The goal is to find the shape of DL so that the focused light in DF is maximized.
Fig. 2
Fig. 2 Left: Grid used to generate multivariate B-splines of degree 3. Right: Multivariate B-splines of degree 3. Its support comprises 4 × 4 grid cells. The B-spline is polynomial in each cell.
Fig. 3
Fig. 3 First numerical experiment: Absolute value of Ez before (a) and after optimization with 49 (b), 289 (c), and 729 (d) multivariate B-splines. The optimized shapes are thicker in order to shift the focus close to the lens surface.
Fig. 4
Fig. 4 Second numerical experiment: Absolute value of Ez before (a) and after (b,c,d) optimization. We observe a high sensitivity of the field distribution.
Fig. 5
Fig. 5 Third numerical experiment: Absolute value of Ez before (a) and after (b,c,d) optimization. The optimized shapes are thinner in order to shift the focus outside the lens. Fourth numerical experiment:Absolute value of Ez before (e) and after (f) optimization for n = 2, and upper half of optimized and initial lens for n = 1.5 (g) and n = 3 (h). For the latter case, we plot |Ez| along the x-axis before (red dotted line) and after (blue line) optimization (i). We observe that the optimum is achieved by drastically increasing transmission.

Equations (13)

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Δ E z k 2 ( x ) E z = 0 in 2 , where k ( x ) : = { n L | k | in D L , | k | in 2 \ D L .
lim | x | | x | ( E z s ( x ) x | x | i | k | E z s ) = 0 .
Ω E z v k 2 E z v d x = Ω E z n v d S , for all v H 1 ( Ω ) ,
J : D L J ( D L ) : = D F | E z | 2 d x .
argmax D L U ad ( D L , 0 ) J ( D L ) subject to Eq . ( 1 ) .
argmax V J ( V ) : = J ( T V ( D L , 0 ) ) subject to
div ( M V grad E z ) k 2 E z det D T V = 0 in 2 ,
V N ( x ) = i = 1 N ( c i 1 c i 2 ) B i ( x ) , c i 1 , c i 2 ,
{ x 3 / 6 , 0 < x 1 , ( 3 x 3 + 12 x 2 12 x + 4 ) / 6 1 < x 2 , ( 3 x 3 24 x 2 + 60 x 44 ) / 6 2 < x 3 , ( 4 x ) 3 / 6 3 < x 4 .
d J ( V , W ) : = { 2 E z h W M V p ¯ h k 2 E z h p ¯ h W ( det D T V ) d x } ,
W M V : = det ( D T V ) ( tr ( D T V 1 D W ) D T V 1 D T V T D T V 1 ( D T V T D W T + D W D T V 1 ) D T V T ) , W ( det D T V ) : = det ( D T V ) tr ( D T V 1 D W ) .
{ div ( M V grad p ¯ ) k 2 p ¯ det D T V = E ¯ z χ D F in 2 , Sommerfeld radiation condition Eq . ( 2 ) ,
( V N update , W N ) H 1 = d J ( V N , W N ) W N as in Eq . ( 7 ) ,

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